Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. (a) -(x + 1)(x + 2) ≥ 0 (b) -(x + 1)(x + 2) > 0 (c) -(x + 1)(x + 2) ≤ 0 (d) -(x + 1)(x + 2) < 0

Verified step by step guidance

1. Start by rewriting the inequality in a clearer form. For example, for part (a), write it as \(- (x + 1)(x + 2) \geq 0\).
2. Identify the critical points by setting the expression inside the inequality equal to zero: solve \(- (x + 1)(x + 2) = 0\). This gives the roots \(x = -1\) and \(x = -2\).
3. Use the critical points to divide the number line into intervals: \((-\infty, -2)\), \((-2, -1)\), and \((-1, \infty)\).
4. Test a sample value from each interval in the original inequality to determine whether the inequality holds in that interval. Remember to consider the effect of the negative sign in front of the product.
5. Based on the test results and the inequality sign (whether it includes equality or not), write the solution set in interval notation, including or excluding the critical points accordingly.

Key Concepts

Factoring and Zero-Product Property

Factoring expresses a quadratic expression as a product of binomials, making it easier to find roots. The zero-product property states that if a product equals zero, at least one factor must be zero. This helps identify critical points where the inequality may change.

Solving Quadratic Inequalities

Solving quadratic inequalities involves determining where the quadratic expression is positive, negative, or zero. This requires analyzing the sign of the expression in intervals defined by the roots, often using test points or sign charts to find solution sets.

Interval Notation

Interval notation is a concise way to represent sets of real numbers between two endpoints. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints), which is essential for expressing solution sets of inequalities clearly.